Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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7
votes
3answers
2k views

Why does the absolute value disappear when taking $e^{\ln|x|}$

I have noticed that if you have an equation (after integrating) such as $$\ln|y| = \ln|x| + c,$$ and you further simplify it using the law of exponents, you get $$e^{\ln|y|} = e^{\ln|x|+с},$$ which is ...
5
votes
2answers
206 views

“Constrained” numerical solutions of ODEs with conservation laws?

Hi know little about numerical methods and I was considering the following problem that possibly has standard solution in the literature. Suppose you have an ODE for wich we already know that it must ...
4
votes
1answer
49 views

what can I say about the solution $y(x)$ of the ODE?

Let $y:\mathbb R\to \mathbb R$ be differentiable and satisfy the ODE: $$\frac{dy}{dx} =f(y),x\in\mathbb R$$ $$y(0)=y(1)=0$$ where $f:\mathbb R\to \mathbb R$ is a Lipschitz continuous function. Then ...
4
votes
2answers
2k views

Relation between Heaviside step function to Dirac Delta function

I understand that "delta function" is a distribution, not a function, as in it acts on another integrand, picking out the value of that integrand at a specific point. The discontinuous function is ...
4
votes
2answers
4k views

How can I solve this Initial Value Problem using the Euler method?

My Problem is this given Initial Value Problem: $$y^{\prime}=\frac{3x-2y}{x}\quad y(1)=0$$ I am looking for a way to solve this problem using the Euler method. I have a given Interval of $[1,2]$ and a ...
4
votes
1answer
184 views

Nonlinear equation (oscillon) comparison

Lagrangian for a spherically-symmetric, real scalar field in d spatial dimensions, $$L=c_d \int r^{d-1}dr\left[ \frac{1}{2} \dot\phi^2 - \frac{1}{2} \left(\frac{\partial \phi}{\partial r} \right)^2 ...
4
votes
2answers
301 views

Formula for integration bounds of recursively defined polynomial sequence

We can recursively define a sequence of polynomials by $$P_0(x) := 1$$ and then with the definite integral $$P_n(x) := \int_{c_n}^x P_{n-1}(t) ~\mathrm dt$$ where the $c_n$ are to be chosen so ...
3
votes
2answers
94 views

Solve $(x^2 + 1)y'' - 6xy' + 10y =0$ using series method

Use series methods to solve: $(x^2 + 1)y'' - 6xy' + 10y =0$ a) Give the recursion formula b) Give the first two non-zero terms of the solution corresponding to $a_0 = 1$ and $a_1 = 0$ ...
3
votes
1answer
2k views

Finding the derivatives of inverse functions at given point of c

Hoping someone can help me the understand the steps to solve a problem like this. I'm guessing it involves the formula: $\frac{d}{dx}f^{-1}(f(x))=1/f'(x)$. Am I right in this assumption? I would post ...
3
votes
2answers
144 views

Solve the pde $u_t(x,t)=u_{xx}(x,t)-bu(x,t)+q(t)$ for $u(x,t)$

I have the example pde $u_t(x,t)=u_{xx}(x,t)-b(t)u(x,t)+q_0$, where $b(t)$ is a function of only $t$ and $q_0$ is a constant, $0<x<\pi$, $t>0$. The subscripts denote derivatives. I also have ...
3
votes
1answer
172 views

Initial value problem $t\frac{dx}{dt}=x+\sqrt{t^2+x^2}$

Another question on ODEs, this time just wondering how I should start with this one. $$t\frac{dx}{dt}=x+\sqrt{t^2+x^2}, \qquad x(1)=0.$$ It looks like a linear ODE and so after playing around with it ...
3
votes
1answer
534 views

Second-order nonlinear ODE with Dirac Delta

Can anyone help me with the following differential equation? $$ 2x(t)x''(t) - x'(t)^2 + kx(t)^2\delta(t - a) =0, $$ where $\delta$ represents the Dirac Delta. I tried Mathematica but with no luck. ...
3
votes
1answer
3k views

Frobenius Method to solve $x(1 - x)y'' - 3xy' - y = 0$

So, Im trying to self-learn method of frobenius, and I would like to ask if someone can explain to me how can we solve the following DE about $ x = 0$ using this method. $$ x(1 - x)y'' - 3xy' - y = 0 ...
3
votes
2answers
237 views

Solving ODE with substitution

I have this as homework: $$(xy^2+y)dx+(x^2y-x)dy=0$$ I tried to solve it by substituting $z=xy+1$, but got the answer like $y=Cxe^{xy}$, which, I guess, is wrong. I tried to solve it couple of ...
3
votes
1answer
515 views

Two-Point boundary value problem

To solve ${d^2y \over dx^2} =f(x)$, $0<x<1$ with $y(0)=\alpha, y(1) = \beta$. We can get a finite difference approximation by taking $$\frac{y_{j+1}-2y_j+y_{j-1}}{h^2} =f_j \\\Rightarrow ...
3
votes
1answer
675 views

Why does acceleration = $v\frac{dv}{dx}$

If we define $x$ = displacement, $v$ = velocity and $a$ = acceleration then I am used to the ideas that $a= \frac{dv}{dt} = \frac{d^2x}{dt^2}$ However I also understand $a=v \frac{dv}{dx}$. Can ...
2
votes
1answer
67 views

Specific system of differential equations

I have the following system of equations: \begin{eqnarray}\frac{dx}{dt} = x(1 - x^2 - y^2) \\ \frac{dy}{dt} = y(4 - x^2 - y^2) \end{eqnarray} I want to prove that if a solutions starts (at time $t = ...
2
votes
2answers
47 views

How to define order of differential equation?

I know how to check order of differential equation but I need to know how to define order of differential equation in words. Please help me.
2
votes
1answer
87 views

Proving that a differential equation has unique solution.

Let $P,Q,f:[-1,1]\to\mathbb{R}$ continuous and $a,b\in\mathbb{R}$. Then I want to prove that the IVP $$\begin{cases} u''(x)+P(x)u'(x)+Q(x)u(x)=f(x)\\u(0)=a,\qquad u'(0)=b\end{cases}$$ has a unique ...
2
votes
3answers
145 views

Numerical estimates for the convergence order of trapezoidal-like Runge-Kutta methods

I have to calculate approximations of the solution with the method $$ y^{n+1}=y^n+h \cdot [\rho \cdot f(t^n,y^n)+(1-\rho) \cdot f(t^{n+1},y^{n+1})] ,\quad n=0,\ldots,N-1 \\ y^0=y_0 $$ for various ...
2
votes
1answer
52 views

Repairing solutions in ODE

Recently I encounter something interesting that I hope to hear from your opinions: Suppose we are given a ODE $\frac{dy}{dx}=y$, with no initial condition. Naively, we divide both sides by $y$ and ...
2
votes
1answer
70 views

Is the continuity of a vector field enough for the existence of the solution of a differential equation?

I've recently seen the existence-uniqueness theorem for ordinary differential equations from Arnold's book. I understand that the theorem as stated guarantees both existence and uniqueness if the ...
2
votes
2answers
169 views

How can I solve these pde's?

Three different problem I got: 1.. $xu_x+2x^2u_y-u=x^2e^x$ and $u(x,x^2+x)=xe^x+x^2$ 2.. $yu_{xx}+(x+y)u_{xy}+xu_{yy}=0, \quad x\neq y$ 3.. $(y+xu)u_x+(x+yu)u_y=u^2-1$ Couldnt even start. Could ...
2
votes
1answer
84 views

Laplace Trouble to find solution

Trying to figure out how to use Laplace Transform to find $y(t)$: The problem is $$y''+4y'+4y=f(t)$$ where $f(t) = \cos(\omega t)$ if $0 < t < \pi$ and $f(t)=0$ if $t > \pi$? Initial ...
2
votes
2answers
80 views

Procedure for 3 by 3 Non homogenous Linear systems (Differential Equations)

Here is the problem I have. $$x^{'}(t)=Ax(t)+f(t)$$ where $A=\begin{pmatrix} 5&-3&-2\\8&-5&-4\\-4&3&3 \end{pmatrix} f(t)=\begin{pmatrix} -\sin (t)\\ 0 \\ 2 \end{pmatrix}$ I am ...
2
votes
1answer
160 views

Finding a value a for topologically conjugacy between two flows

Let A be a hyperbolic matrix such that all solutions of $\overrightarrow x' = A \overrightarrow x $ tend to the origin at t goes to infinity, and suppose B = $\begin{bmatrix}a-3 & 5 \\ -2 & ...
2
votes
1answer
951 views

Solving a partial differential equation using method of characteristics

I keep getting stuck and have a hard time understanding my professor, so I'm hoping to get some help here. The question is: Solve the partial diff/eq: ${\partial u}\over{\partial t}$ + $c {{\partial ...
2
votes
1answer
223 views

Eigenvectors Trajectories

I got stuck with a problem while studying for a control systems exam. It goes as following: "Look at the picture of trajectories of a linear, time-invariant system with the form: ...
2
votes
2answers
70 views

second order DE using reduction of order

Any Hints / details on how to find a second solution for $$x^2y'' + xy' -4y=0?$$ $$y_1 = x^2 y_2$$ I need to use reduction of order thanks
2
votes
1answer
133 views

Choice of the First Term in Legendre Polynomials

The two solutions of the Legendre's Differential Equation obtained by series solution method are : and Now according to my textbook, for the useful polynomial for n equal to a positive integer, ...
2
votes
0answers
838 views

Hard Differential Equation. Please help.

first of all I'm not a mathematician, so I apologize if any of my understanding and terminology isn't up to par. Also, I've never used this website (or any of these kind of question/answer) websites ...
2
votes
0answers
152 views

Calculate half life of esters

I'm trying to calculate the level of testosterone released from different testosterone esters. Here are some graphs of testosterone levels after single injections of 250mg of each ester. Testo U ...
2
votes
1answer
3k views

How can I use the Heun's method to solve this first order Initial Value Problem?

My Problem is this given Initial Value Problem: $$y^{\prime}=\frac{3x-2y}{x}\quad y(1)=0$$ I am looking for a way to solve this problem using Heun's method. I have a given Interval of $[1,2]$ and a ...
2
votes
3answers
831 views

Express differential equations as system of first order equations

Express the differential equation $$y'''-6y''-y'+6y=0$$ as a system of first order equations i.e. a matrix equation of the form $$A(\vec x)'=0$$ where $$\vec x\text{ is the vector }\left[ ...
2
votes
2answers
609 views

Considering the linear system $Y'=AY$

What would be an equation that I can use when I compute the eigenpairs for the coefficient matrix $A.$
2
votes
0answers
71 views

Differential Equation - $y'=5|y|^{4/5}, y(0)=0$

in the spirit of this question I ask about this one. $y'=5|y|^{4/5}, y(0)=0$ If $y> 0$ then $$y'=5|y|^{4/5}\iff y'=5^{-1}y^{4/5}\iff 5^{-1}y'y^{-4/5}=1\iff y^{1/5}=x+C\\ \iff ...
2
votes
1answer
66 views

Differential Equation - $y'=|y|+1, y(0)=0$

The equation is $y'=|y|+1, y(0)=0$. Suppose $y$ is a solution on an interval $I$. Let $x\in I$. If $y(x)\ge 0$ then $$y'(x)=|y(x)|+1\iff y'(x)=y(x)+1\iff \frac{y'(x)}{y(x)+1}=1\\ \iff \ln ...
2
votes
1answer
641 views

butcher tableau runge kutta methods

Hi I have had a go at this question- am i heading in the right direction? it would be much appreciated if someone could me Write the Butcher Tableau for the 1-stage $\theta$ method: $$U^n ...
2
votes
2answers
637 views

Generalized “Worm on the rubber band ” problem

I found this « Worm on the rubber band » problem in Concrete Mathematics book. A slow worm $W$ starts at one end of a meter-long rubber band and crawls one centimetre per minute toward the other end. ...
1
vote
2answers
53 views

find an approximate solution, up to the order of epsilon

The question is to find an approximate solution, up to the order of epsilon of following problem. $$y'' + y+\epsilon y^3 = 0$$ $$y(0) = a$$ $$y'(0) = 0$$ I tried to solve the given problem using ...
1
vote
1answer
90 views

$y(x)$ be a continuous solution of the initial value problem $y'+2y=f(x)$ , $y(0)=0$

Let , $y(x)$ be a continuous solution of the initial value problem $y'+2y=f(x)$ , $y(0)=0$ , where, $$f(x)=\begin{cases}1 & \text{ if } 0\le x\le 1\\0 & \text{ if } ...
1
vote
2answers
58 views

Seemingly easy Ordinary Differential Equation

For which values of $T$ can we find a unique solution of the ODE $x''(t) = −x(t) $ satisfying the boundary conditions $x(0) = a_1$ and $x(T) = a_2$ for any values of $a_1$ and $a_2$ ? I can solve ...
1
vote
1answer
191 views

How to find the order of accuracy of this implicit RK method (using Taylor series)?

I want to get the order of accuracy (local truncation error - LTE) of this implicit 2-step method. The first step is Backward Euler to determine an approximation to the value at the midpoint in time, ...
1
vote
2answers
95 views

Numerical Approximation of Differential Equations with Midpoint Method

I want to proof that the local truncation error of the Midpoint Method is $d_{k+1}=O\left(h^{3}\right)$ Approach The local truncation error is defined as: ...
1
vote
1answer
64 views

Modelling population with $\frac{dP}{dt}=P(\beta - \delta P)$

The population $P(t)$ of a biological species can be modelled by $$ \frac{dP}{dt}=P(\beta - \delta P) $$ subject to $P(0)=P_0$ where $\beta$ is the birth rate and $\delta$ is the death rate. ...
1
vote
2answers
49 views

Differentiation - simple case

In the book calculus made easy, page 22 the case of the negative power for $y=x^{-2}$ $$\begin{align} y+dy & =(x+dx)^{-2}\tag{1}\\ \\ & = x^{-2}\left(1+\frac{dx}{x}\right)^{-2}\tag{2} ...
1
vote
3answers
306 views

Tricky Separable Differential Equation

Please guide me: $y' + ay +b = 0$ (a not zero) is supposed to be separable and has solution $y = ce^{-ax} - \frac ba$ Here is my start to this problem: $\frac{dy}{dx} + ay = -b$ is as far as I can ...
1
vote
2answers
336 views

Proving a function is Lipschitz continuous

Show that the following function is Lipschitz continuous and find a Lipschitz constant $$y\mapsto f(x,y)\\ f(x,y)=\frac{y}{x}\ln(\frac{y}{x})\text{ , } |x-1|\leq\frac{1}{2}\text{ , } ...
1
vote
2answers
130 views

How to solve this differential equation system?

The following system is given: $$ \dot{x} = y + z \\ \dot{y} = x + z \\ \dot{z} = x + y $$ The first thing I did was to find out the eigenvalues. I found out, that -1 is a doubled and 2 a single ...
1
vote
0answers
167 views

Differential equation with random variable

How can I derive analytically or compute numerically the solution to following differential equation $$ dy/dt = y\cdot X\cdot (y\cdot X - g(y,X))\cdot X $$ where X is a random variable (e.g. from a ...